“It should be noted that even when the correlation is as high as .90, there will be some individuals very high in one ability and very low in the other. Such disparities are to some extent, as Courtis and Cobb have argued, due to inborn characteristics of the individual in question, which predispose him to very special sorts of strength and weakness. They are often due, however, to defects in his learning, whereby he has acquired more ability than he needs in one line of work, or has failed to acquire some needed ability, which was well within his capacity.
“In general, all correlations between an individual’s divergence from the common type or average of his age for one arithmetical function, and his divergence from the average for any other arithmetical function, are positive. The correlation due to original capacity more than counterbalances the effects that robbing Peter to pay Paul may have.”
In 1910, Brown undertook to determine whether there is a special capacity for mathematics, and concluded from his correlations that there is an especially close relationship among tests involving mathematical performance. Ten years later, Collar made an effort to secure further data as to whether arithmetical ability, as a unitary combination of capacities, exists. Two hundred schoolboys were tested in the investigation. Results led to the conclusion that arithmetical ability tends to be represented in two main divisions: (1) the power to compute with ease and readiness, and (2) the power to solve problems by arithmetic, which involves the application of a higher degree of ability than is required in computation.
Arithmetical tests of various kinds correlate more closely than do arithmetical tests with non-arithmetical tests. “Hence we are compelled to interpret this relationship as evidence distinctly in favor of Burt’s suggestion, that there is an essential unity in arithmetical ability.”
All investigators have agreed in finding the correspondence between computation and problem-solving much less than that found among the various processes of computation alone. The facts are here analogous to certain facts noted in the study of reading, in Chapter IV. There it was seen that between proficiency in the mechanics of reading and comprehension in reading there may occur marked disparity; and that it is in mechanics that special discrepancies may be found between reading ability and general intelligence.
In arithmetic the same observation may be made. Marked special defects and talents are found in the mechanics of arithmetic, that is, in computation. But problem-solving in arithmetic is closely correlated with general intelligence, for it involves the capacities required for problem-solving anywhere,—response to many subtle elements, the weighing of these one against another, and choice of the procedure that will yield solution. These are the same capacities that underlie comprehension in reading, or grasp of any other situation offered by life. They are all functions measured in tests of general intelligence.
In school, arithmetical problems are usually presented as reading matter, so that reading for the comprehension of sentences is in itself of first rate importance for achievement in problem-solving.
V. PSYCHOLOGICAL STUDIES OF SPECIAL DEFICIENCY IN ARITHMETIC
Studies of children especially backward in arithmetic, with the accounts of the results of experimental teaching, have been contributed by Uhl, Smith, Schmitt, and others. Bronner has also contributed accounts of the psychological examination of such children.
Schmitt studied thirty-four pupils in the schools of Chicago, who were not feeble-minded, but were extremely retarded in arithmetic. The investigator states that tests of general intelligence were given, but does not share with the reader the exact results of such tests, saying only that the children “were not mentally defective.” The result of tabulation of circumstances involved showed that ill-health and absence were closely related to special disability in arithmetic. The inference is drawn that achievement in arithmetic calls for a hierarchy of habits, which depend on each other in a sequence. If a hiatus occurs at any essential point, as through absence, inattention, or inadequate teaching, confusion follows. (This inference seems very well justified, also, from the psychological analysis of the mental functions involved in arithmetic.) The problem of individual examination is to find out what habits have not been formed. The problem of pedagogy is to teach those habits, and to motivate the child.